High-surface-area corundum nanoparticles by resistive hotspot-induced phase transformation

High-surface-area α-Al2O3 nanoparticles are used in high-strength ceramics and stable catalyst supports. The production of α-Al2O3 by phase transformation from γ-Al2O3 is hampered by a high activation energy barrier, which usually requires extended high-temperature annealing (~1500 K, > 10 h) and suffers from aggregation. Here, we report the synthesis of dehydrated α-Al2O3 nanoparticles (phase purity ~100%, particle size ~23 nm, surface area ~65 m2 g−1) by a pulsed direct current Joule heating of γ-Al2O3. The phase transformation is completed at a reduced bulk temperature and duration (~573 K, < 1 s) via an intermediate δʹ-Al2O3 phase. Numerical simulations reveal the resistive hotspot-induced local heating in the pulsed current process enables the rapid transformation. Theoretical calculations show the topotactic transition (from γ- to δʹ- to α-Al2O3) is driven by their surface energy differences. The α-Al2O3 nanoparticles are sintered to nanograined ceramics with hardness superior to commercial alumina and approaching that of sapphire.


Supplementary Note 1. Crystalline size determination by Halder-Wagner's method.
The diffraction peak shows a visible spread if the crystalline size is smaller than 100 nm or if lattice strain is present. Hence, the diffraction peak spread is used to analyze crystalline size and lattice strain. Prior to analysis, the diffraction peak spread is corrected by the X-ray diffractometer.
According to the Halder-Wagner's method in Supplementary Equation (1) (ref. 1 ), 2 tan 2 = tan •sin + 16 2 (1) where θ is the diffraction angle, K is the Scherrer constant (being 1), L is the crystalline size, λ is the wavelength of wavelength of the X-ray (being 1.5406 Å here for Cu Kα), e is the lattice strain, and β is the integral width of corresponding diffraction peak and determined by Supplementary Equation (2), where S is the integral width of a Lorentzian function which is described as the crystalline size, and D is the integral width of a Lorentzian function which is described as the lattice strain.
As shown in Supplementary Fig. 12, we plotted 2 tan 2 against tan •sin and obtained the crystalline size L ~22 nm and lattice strain e = 0% of the samples.
Supplementary Note 2. The simulation of current density.

Volume fraction of γ-Al2O3 NPs.
The volume fractions (f) of the γ-Al2O3 were calculated based on the mass ratio and the densities

Simulation method.
The simulation was conducted based on the finite element method (FEM) by using the COMSOL Multiphysics 5.5 software. The Electric Currents interface in AC/DC module was used as the model. To simplify the simulation, we used a two-dimensional configuration. The geometric configuration and materials parameters were shown in Supplementary Table 3 and Supplementary where V0 is the overall potential (60 V), L0 is the length of the sample (5 mm), and L is the geometrical size for the simulation. As a demonstration of f(γ-Al2O3) = 0.42, the γ-Al2O3 NPs were square packed, and the boundary conditions were shown in Supplementary Fig. 15a. The simulated electric potential map ( Supplementary Fig. 15b) showed the linear decrease of electric potential from the Electric Potential boundary to Ground boundary. The simulated current density maps for various volume fraction of γ-Al2O3 are shown in Supplementary Fig. 16.

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We focused on the volume fraction of f(γ-Al2O3) = 0.73 (Fig. 3e), under which condition the phase transformation takes place most rapidly and thoroughly. The generated heat is determined by the Joule heating equation, where Q is the heat amount, I is the current, R is the resistance, and t is the heating time.
On the other hand, the temperature change is determined by the following equation, where Q is the heat amount, c is the heat capacity of the sample, m is the sample mass, and ΔT is the temperature change. According to Supplementary Equations (5-6), we get the Supplementary Equation (7), Since c, m, R, and t are constant values for a specific sample, the temperature change is proportional to the square of current. For simplicity, we use the temperature changes and current per volume to revise Supplementary Equation (7), and we got the following equation, where j is the current density. The average bulk temperature could be experimentally measured (Tbulk = 573 K, Supplementary Fig. 14), which corresponds to the average bulk current density (jbulk). The current density distribution contour map is obtained by numerical simulation ( Supplementary Fig. 18a). We only consider the current passing through the carbon black since the Al2O3 phase is insulative and thus its current density is zero. Hence, from the contour map, the average bulk current density was calculated to be jbulk = 4.4 × 10 6 A m -2 .
As the name suggests, "hotspot" is a region wherein the temperature is substantially higher than the average bulk temperature. Hence defining a specific "hotspot zone size" is somewhat S5 blurred. Nevertheless, it is reasonable to define a threshold that the temperature is higher than the  Supplementary Fig. 18b). In the hotspot zone, the temperature is Thotspot ≥ 1473 K, and even higher at the center part of the hotspot zone. This high temperature triggers the ultrafast phase transformation.
According to Supplementary Fig. 18b, all particles were heated to above the phase transformation temperature at two vertical gaps regions. Furthermore, for a specific particle, ~29% of its surface area was heated to above the phase transformation temperature. This also suggests that phase transformation would take place at the surface and then penetrate into the bulk of the particle.

The effect of geometric configuration.
To demonstrate the effect of geometric configuration, we also simulated the hexagonal packing order using the same protocols ( Supplementary Fig. 17). The current maps also show local maximums, or hotspots, in the vertical gaps between γ-Al2O3 NPs. These results show that the hotspot effect is just affected by the gap dimension of the γ-Al2O3 NPs and not related to the geometrical configuration. In the real sample, we used ball milling to mix γ-Al2O3 NPs and CB to ensure a best mixture. The real configuration of γ-Al2O3 NPs in CB matrix is three dimensional and random. Nevertheless, from the simulation results of the square stacking and hexagonal S6 stacking, it is reasonable to conclude that the proposed hotspot effect is available in the real threedimensional system.

Supplementary Note 3. Consideration of surface OH group on γ-Al 2 O 3 .
The surface energy of the pristine and OH-adsorbing Al2O3 surfaces are plotted in Supplementary Fig. 20 with respect to OH coverage. Notice that in Supplementary Fig. 20, the surface energy decreases linearly with the increase of OH coverage. This allows for a fit of surface energies at 2 OH nm -2 , which are shown in Supplementary Table 6. Also, it is important to see that the surfaces energy of the γ-Al2O3 surfaces are significantly lower than α-Al2O3 and δʹ-Al2O3. This is why at smaller sizes, the nanoparticles of γ-Al2O3 are more stable. Furthermore, the surface energy of γ-Al2O3 surfaces decrease much faster with OH adsorption, which explains the fact that we only observed OH adsorption on γ-Al2O3 NPs ( Supplementary Fig. 19). In contrast, there is barely any OH on the α-Al2O3 and δʹ-Al2O3 NPs.
The optimized atomic structures of γ-Al2O3 with surface OH groups density of ~2 OH nm -1 are shown in Supplementary Fig. 21.
Supplementary Note 4. Nanoparticle shape optimization by Wulff theorem.

Wulff energy for nanocrystals and surface energy of Al2O3.
Optimization of the shape of nanocrystals is based on the generalized Wulff theorem 3 , where f is the total energy of the particle, μ is the bulk energy per formular unit (or atom), N is the S7 total number of formular unit in the particle, Si is the area of the i th facet of the nanocrystal, is the surface energy of the i th facet of the nanocrystal, is the length of the j th edge, is the edge energy, and is the energy of the k th vertex. The above equation cannot be rigorously treated if the shape is complicated with all possible facets considered. The equation can be further simplified by considering only the most probable facets with relatively lower indexes because the higherindex facets normally have much higher energy or complicated reconstruction, which make them unlikely to occur in nanocrystals. Therefore, in this study, we only consider three types of facets for the Al2O3 nanocrystals, equivalent to the surfaces listed in Supplementary Table 6, which are normally considered in this field 4 .
Considering that (112 ̅ 0) and (11 ̅ 00) surfaces form 150º angle, and they are orthogonal to (0001) surface, these surfaces naturally form a dodecagonal prism with bottom facets being (0001) surfaces and the side facets being (112 ̅ 0) and (11 ̅ 00) surfaces arranged alternatively side by side ( Supplementary Fig. 24a). Because the surface energies of (112 ̅ 0) and (11 ̅ 00) are very close, we treated them equally in optimization. Assuming that the radius of circumcircle of the bottom dodecagon is x and the height of the dodecagonal prism is a, the volume of the prism V, the area of bottom facets S0, and the area of the side facets S1 can be expressed as,

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The reduced Wulff energy (the bulk energy is a constant) is expressed as, ( ) = 0 0 + 1 1 = 6 0 2 + 8 1 where 0 is the surface energy of (0001), and 1 is the average energy of (112 ̅ 0) and (11 ̅ 00) surface. Therefore, the above function is minimized at, With this structural parameter, one can easily find the relationship between the specific surface area A of the nanocrystals and the Wulff energy of α-Al2O3 nanocrystals ( ), where the atomic mass Al = 26.9815, O = 15.9994, atomic unit of mass u = 1.6605 × 10 -24 g.
The bulk energy, surface energies, and density n for α-Al2O3 are given in Supplementary Table 6.
According to Supplementary Equations (10)(11)(12), and (14), for a particular V, there is a unique A and α . Therefore, we can plot α as the function of A .

Optimized shape of γ-Al2O3 nanocrystals.
The energy of (100) surface of γ-Al2O3 is significantly lower than that of the (110) surface.
Therefore, (100) facets will dominate in the formation of nanocrystals. According to our derivation, (110) facets can only coexist with (100) when its surface energy is lower than √2 of that of (100). This condition is not satisfied according to the values listed in Supplementary Table   6. That means that the (110) facets cannot occur. Since the (111) surface has a lower energy than the (110) but higher than (100), we consider truncated cubic shape of γ-Al2O3 nanocrystals with six square (100) facets and eight triangular (111) facets. Assuming that the edge of the cube is a S9 and a portion of x ( ≤ 2 ) along the edge is truncated off at each corner ( Supplementary Fig. 24b), the volume V, the total area of the (100) facets 0 , and the total area of the (111) facets 1 , are expressed respectively as, The reduced Wulff energy (the bulk energy is a constant) is where 0 and 1 are the surfaces energy of (100) and (111), respectively. The above energy is minimized at, With this structural parameter, one can similarly find the relationship between the specific surface area A of the nanocrystals and the Wulff energy of γ-Al2O3 nanocrystals ( γ ), The bulk energy γ , surface energies of (100) and (111) facets, and density n for γ-Al2O3 are given in Supplementary Table 6. According to Supplementary Equations (17)(18)(19), and (21), for a particular V, there is a unique A and γ . Therefore, we can plot γ as the function of A .

Optimized shape of δʹ-Al2O3 nanocrystals
We note that in the truncated cube case of γ-Al2O3 nanocrystals, the truncated portion of the edge x must be less than half of the length of the edge a. When the energy of (111) surface is further lowered than (100) surface, the optimized = (3 0 − √3 1 ) /2 0 can be larger than a/2.
Then the shape of the nanocrystals transforms into truncated octahedron with six square (100) facets and eight hexagonal (111) facets. This is the case of δʹ-Al2O3 nanocrystals. Assuming that the edge of the octahedron is a and a portion of x along the edge is truncated off at each corner ( Supplementary Fig. 24c), the volume V, the total area of the (100) facets 0 , and the total area of the (111) facets 1 , are expressed respectively as, The reduced Wulff energy (the bulk energy is a constant) is expressed as, where 0 and 1 are the surfaces energy of (100) and (111), respectively. When minimizing the reduced Wulff energy, one finds the structure parameter of the truncated octahedron at, Again, we obtain the specific surface area A of the nanocrystals and the Wulff energy of δʹ-Al2O3 nanocrystals ( δʹ ), δʹ = δʹ + 0 0 + 1 1 = δʹ + 0 0 + 1 1 .
The bulk energy δʹ , surface energies of (100) and (111), and density n for δʹ-Al2O3 are given in Supplementary Table 6. According to Supplementary Equations (24-26), and (30), for a particular V, there is a unique A and δʹ . Therefore, we can plot δʹ as the function of A . S11

Supplementary Note 5. The entropic contribution.
For solid reactions, the entropy contribution to the free energy change is usually much smaller than the enthalpy contribution. Hence, in our calculation, we only considered the energy (including bulk energy and surface energy) of the three phases of Al2O3.
We here estimated the entropic contribution to the phase transformation. According to the NIST table 5 , under standard conditions, the entropies of three Al2O3 phases are (we note that entropy of δʹ-Al2O3 could not be found in literature; since the δ-Al2O3 and δʹ-Al2O3 phases have similar crystal structures, we here used the entropy of δ-Al2O3 as an alternative, These entropy values could be converted to those per Al2O3 unit, These values are about ~1% of the enthalpy values of the three phases, indicating that the entropy contribution is much less than the enthalpy contribution to the free energy.
According to the Gibbs free energy equation, ∆ = ∆ − ∆ , the free energy of the three phases could be calculated. We plot the free energy and specific surface area, as shown in Supplementary Fig. 25. The main conclusion remains the same when just consideration of enthalpy, that is, the surface energy difference between the three phases drives the phase transformation from γ-Al2O3 to δʹ-Al2O3 and then to α-Al2O3 phase. The transformative surface area from δʹ-Al2O3 to α-Al2O3 phases happens at ~96 m 2 g -1 , comparable to the value of ~93 m 2 g -1 that just considering the energy calculation.

Supplementary Note 6. Dynamic simulations of the structural transformation.
In this study, we conduct dynamic simulations of phase transition of nanocrystals at two critical sizes, i.e., α-to γphase transition at <2 nm and γ-to αphase transition at >20 nm. The smaller size (α-to γ-phase) can be directly handled in DFT simulation. But direct MD simulation of the larger particles (γ-to αphase) using the DFT method is not feasible. To this end, we performed MD modeling of bulk phase transition using the Langevin thermostat in the isobari-S13 isothermic NPT ensemble 6,7 , which allows for transformation of symmetry and size of the supercell. Although the initial crystalline structure can be chosen, the final structures formed are normally amorphous due to the limited time scale of simulation (<100 ps) and, for the bulk phase transition, the incommensurate lattice at the limited size of the supercell. Therefore, alternatively, We first performed NVT dynamics simulation of a 1.2×1.5×1.8 nm 3 α-phase nanocrystal at 1800 K for 10 ps following a 2 ps preheating. The hexagonal structure is transformed into an amorphous structure in less than 5 ps (Supplementary Figs. 26a-b), demonstrating the structural instability of α-phase at very fine particle size. After 10 ps, the system is quenched to 100 K in 2 ps and followed by structure optimization (Supplementary Fig. 26c). On the surface of the particle, the rectangular local bonding network is recognized (circles in Supplementary Fig. 26d); when looking into the particle, a substantial amount of tetrahedral Al atoms can be seen (arrows in Supplementary Fig. 26e). Both are typical features of local order of the γ-phase.
We further performed dynamic simulations of a 1.8-nm slab of the α-phase with both sides being the (112 ̅ 0) surface. The simulation time is set to 20 ps, while the temperature varies from S14 to 20 ps. The optimized final structures show that four surface layers of atoms were transformed ( Supplementary Fig. 27c). Similar structural transformation takes places irreversibly at all temperatures, indicating that these surface structures are energetically more favorable than the original (112 ̅ 0) surface of α-phase. Again, we can recognize the local rectangular order on the surface (circles in Supplementary Fig. 27d), and the tetrahedral bonding configuration of some Al atoms in the deeper layer (arrows in Supplementary Fig. 27e). Therefore, based on the dynamic simulations, we demonstrate the high surface energy of the α-phase drives the structural transformation, consistent with the energy diagrams (Fig. 4b).
When the nanocrystals are big enough, the bulk energy lowering dominates the energy landscape, so the γ-phase would transform into the α-phase. To verify this, we constructed a γ- which features the local order of α-Al2O3, demonstrating the bulk densification. Therefore, based on the dynamic simulation, we demonstrate the high bulk energy of γ-phase drives the structural transformation, consistent with the energy diagrams (Fig. 4b). S15

Supplementary Note 7. Electrical energy consumption for the phase transformation.
The energy consumption was calculated by Supplementary Equation (31), where E is the energy per gram (kJ g -1 ), V0 and V1 are the start voltage and voltage after Joule heating, respectively, C is the capacitance (C = 0.624 F), and M is the mass of Al2O3 per batch.
Given that the industrial price of electric energy in Texas, USA is $0.02/kWh, the electrical energy cost of the synthesis of corundum nanoparticles would be P = 0.027 $ kg -1 .
We compared our process with the other thermal processes with regard to energy consumption for the phase transformation synthesis of corundum NPs. We considered a furnace annealing process (1473 K and 10 h) 2 by using a commercial Muffle furnace (KSL-1200X-UL, MTI) with mass loading of ~10 kg and power of 3 kW. The energy consumption is calculated to be ~108 kJ g -1 . Hence, the resistive hotspot enabled localized heating in PDC process is 20× less energy consumptive than a normal thermal process. In the PDC process, the temperature is the key thermodynamical handle for the phase transformation of the γ-Al2O3. Hence, to maintain a constant temperature values and distribution is critical for the scaling up of the PDC process.

Parameters determining the current density.
We where S is the cross-sectional area of the sample.
The resistance of the sample is determined by Supplementary Equation (35),

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where ρe is the resistivity of the sample, and L is the length of the sample.

The sample mass (m) is calculated by Supplementary Equation (36),
where ρm is the density of the sample.
Above all, we get the expression of the current density by Supplementary Equation (37), Since ρm and ρe are constant for a specific f(γ-Al2O3), to maintain a constant current density when increasing the mass, we need to increase either the sample cross sectional area or the voltage proportionally. Practically, increasing the voltage is feasible. High voltage (>1 kV) or even ultrahigh voltage (hundreds of kV) technologies are well-established.

Scaling up to gram scale.
In this work, most of the synthesis are conducted by using a sample mass of m0 ~200 mg, tube diameter of D0 = 8 mm, and V0 = 60 V. To scale up the synthesis, we used a tube diameter of D1 = 16 mm ( Supplementary Fig. 37). Hence, the cross-sectional area ratio will be S1/S0 Rwp value of ~4.57% demonstrates a good convergence of the refinement.

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Discussion: The refinement shows that the starting materials are composed of ~91 wt% γ-Al2O3 phase and ~9 wt% γ-Al(OH)3 phase ( Supplementary Fig. 6a). The crystalline size of the γ-Al2O3 phase is estimated to be ~4 nm, and γ-Al(OH)3 being ~23 nm. No other phase (e.g., δ-Al2O3 phase) is identified. The γ-Al(OH)3 phase is easily to be converted to γ-Al2O3 by a mild condition calcination ( Supplementary Fig. 9), or by 0.3-s PDC Joule heating (Fig. 1c). We then conducted the Rietveld refinement of the γ-Al2O3 phase obtained by calcination in air at 700 °C for 1 h ( Supplementary Fig. 6b). The refinement shows the pure phase of γ-Al2O3. No other phase (e.g., δ-Al2O3 phase) is identified. The crystalline size of the γ-Al2O3 remains the same of ~4 nm. We note that the conversion of γ-Al(OH)3 to γ-Al2O3 happens before the phase transformation of γto δʹ-Al2O3 phase. Hence, we will not consider the phase of γ-Al(OH)3 when explaining the phase transformation pathway from γ-to δʹand then to α-Al2O3 phase. The XRD pattern of the γ-Al2O3 NPs precursors (Fig. 1c, 0 s) shows that the composition of the precursor is mainly γ-Al2O3 phase with minor content of γ-Al(OH)3. After calcination (in air, 700 °C for 1 h), the XRD pattern ( Supplementary Fig. 9a) shows the disappearance of γ-Al(OH)3 and the retention of γ-Al2O3 phase. No other phase is detected. The surface area and pore width of the calcined samples are ~156 m 2 g -1 and ~5.9 nm, respectively, which are comparable to the values of the γ-Al2O3 precursors at ~156 m 2 g -1 and ~5.6 nm, respectively ( Supplementary Fig. 5). These results show that the calcination process, 700 °C in air for 1 h, does not trigger the phase transformation and has negligible effect on the coarsening or aggregation of the γ-Al2O3 phase.
The HRTEM images of the α-Al2O3 NPs in Supplementary Fig. 10a show the particle size of 20-30 nm. The single set of lattice fringe (blue lines in Supplementary Figs. 10a-c) across the entire particle demonstrates the single-crystal feature of the particle. The single set of diffraction pattern by NBD also explicitly proves the single-crystal feature of the NP (Supplementary Fig. 10d). We can observe the surface morphology feature on the nanoparticles (red circles in Supplementary   Fig. 10a), which is presumed to be retained from the particle fusing of the of γ-Al2O3 NPs precursors. During the PDC Joule heating process, the γ-Al2O3 NPs undergo both phase transformation and grain growth. The two processes inevitably happen simultaneously upon heating. One γ-Al2O3 NP undergoes phase transformation and nucleates as α-Al2O3 NP ( Supplementary Fig. 10e, middle). At the same time, the α-Al2O3 NP nucleus consumes a few nearby γ-Al2O3 NPs and becomes a larger particle ( Supplementary Fig. 10e, right). During the process, the morphology feature of the γ-Al2O3 NP is retained (red dash circles in Supplementary   Fig. 10e, right). We note that the grain growth is inevitable for any thermal process. For the conventional extended high-temperature thermal annealing process, like furnace annealing, the grain coarsening is more detrimental, so the resulting α-Al2O3 has a surface area well below 10 m 2 g -1 . In contrast, since our PDC process is ultrafast and the nanoparticles are locally heated due to the hotspot effect, the mass transfer and grain coarsening during the phase transformation process are, to a large extent, avoided and hence a high surface area of ~65 m 2 g -1 is maintained. The hydrodynamic diameter distribution shows the maximum probable diameter (dmax) at 46 nm.
This value is larger than the particle size from the TEM statistic (~23 nm), which is originated from the different measurement method and consistent with the previous report 4 . There is a tail at the larger diameter side, which might be from some aggregate of the particles. The measurement shows that the α-Al2O3 NPs are dispersible. In addition to the absolute value of the current density in the hotspot, the zone size of the hotspot region should also be considered. Although the sample with f(γ-Al2O3) = 0.78 has the largest current density, the current density map shows that the hotspot region is much smaller than other samples ( Supplementary Fig. 16). In this case, it could not trigger the phase transformation. Discussion: we measured the hydroxyl group density on the γ-Al2O3 surface by TGA ( Supplementary Fig. 21b). The OH groups could chemically adsorb on the surface of γ-Al2O3 at room temperature. The phase transformation from γ-Al2O3 to δʹ-Al2O3 usually initiates at ~700 °C (ref. 9 ); hence, it is reasonable to consider the surface OH group density of γ-Al2O3 at 700 °C to avoid overestimation. The weight loss was measured to be ~1% from 700 °C to 1100 °C, which was resulted by the loss of H2O. The OH density on the γ-Al2O3 surface was calculated to be ~2 OH nm -2 according to its surface area of ~156 m 2 g -1 (Supplementary Fig. 9). The α-Al2O3 green body were put in between the carbon papers, which were connected to the ACS system.  Supplementary Table 10.

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Discussion: (1) The green bodies from different precursors have almost the same densities, demonstrating that the hydraulic press process is similar for two precursors under the same conditions.
(2) For both the newly developed ACS process and the TS-PS process, the density of the ceramics from the as-synthesized α-Al2O3 NPs is higher than that from the commercial α-Al2O3 nanopowders, indicating that the fine nanoparticle precursors facilitate the densification process.
(3) The density of the ceramic sintered by TS-PS process is somewhat higher than that by ACS process. Nevertheless, it is already promising that the 1 min ACS process could achieve a density of ~97% for the as-synthesized α-Al2O3 NPs, once again demonstrating that the fine nanoparticles as precursors is beneficial for the fast deification. Discussion: By using the ACS process, the ceramics from the as-synthesized α-Al2O3 NPs precursor has a fine average grain size (~0.12 μm), much smaller than that from the commercial α-Al2O3 nanopowders (~1.15 μm).